Why is this representation of $C^*\!$-algebras irreducible?

Question

I'm focused on the text "Invitation to $C^*\!$-algebras" by Arveson. In this notation $\mathcal{C}(\mathcal{H})$ is the set of compact operators on $\mathcal{B}(\mathcal{H})$. Here is the relevant text:

Question: Why is $\alpha$ an irreducible representation? I can see it is representation. Maybe the result that the only closed ideals in $\mathcal{C}(\mathcal{H})$ are trivial is relevant here.

Thanks!

Answer 1

Irreducible means that $\alpha(\mathcal C(\mathcal H))'=\mathbb C\,I$. And the commutant of $\mathcal C(\mathcal H))$ is trivial.